DOCSLIB.ORG

8.6 Modular Arithmetic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
8.6 Modular Arithmetic “mcs” — 2015/5/18 — 1:43 — page 263 — #271 8.6 Modular Arithmetic On the first page of his masterpiece on number theory, Disquisitiones Arithmeticae, Gauss introduced the notion of “congruence.” Now, Gauss is another guy who managed to cough up a half-decent idea every now and then, so let’s take a look at this one. Gauss said that a is congruent to b modulo n iff n .a b/. This is j written a b.mod n/: ⌘ For example: 29 15 .mod 7/ because 7 .29 15/: ⌘ j It’s not useful to allow a modulus n 1, and so we will assume from now on that moduli are greater than 1. There is a close connection between congruences and remainders: Lemma 8.6.1 (Remainder). a b.mod n/ iff rem.a; n/ rem.b; n/: ⌘ D Proof. By the Division Theorem 8.1.4, there exist unique pairs of integers q1;r1 and q2;r2 such that: a q1n r1 D C b q2n r2; D C “mcs” — 2015/5/18 — 1:43 — page 264 — #272 264 Chapter 8 Number Theory where r1;r2 Œ0::n/. Subtracting the second equation from the first gives: 2 a b .q1 q2/n .r1 r2/; D C where r1 r2 is in the interval . n; n/. Now a b.mod n/ if and only if n ⌘ divides the left side of this equation. This is true if and only if n divides the right side, which holds if and only if r1 r2 is a multiple of n. But the only multiple of n in . n; n/ is 0, so r1 r2 must in fact equal 0, that is, when r1 rem.a; n/ WWD D r2 rem.b; n/. ⌅ WWD So we can also see that 29 15 .mod 7/ because rem.29; 7/ 1 rem.15; 7/: ⌘ D D Notice that even though “(mod 7)” appears on the end, the symbol isn’t any more ⌘ strongly associated with the 15 than with the 29. It would probably be clearer to write 29 7 15, for example, but the notation with the modulus at the end is ⌘mod firmly entrenched, and we’ll just live with it. The Remainder Lemma 8.6.1 explains why the congruence relation has proper- ties like an equality relation. In particular, the following properties7 follow imme- diately: Lemma 8.6.2. a a.mod n/ (reflexivity) ⌘ a b IFF b a.mod n/ (symmetry) ⌘ ⌘ .a b AND b c/ IMPLIES a c.mod n/ (transitivity) ⌘ ⌘ ⌘ We’ll make frequent use of another immediate corollary of the Remainder Lemma 8.6.1: Corollary 8.6.3. a rem.a; n/ .mod n/ ⌘ Still another way to think about congruence modulo n is that it defines a partition of the integers into n sets so that congruent numbers are all in the same set. For example, suppose that we’re working modulo 3. Then we can partition the integers into 3 sets as follows: :::; 6; 3; 0; 3; 6; 9; : : : f g :::; 5; 2; 1; 4; 7; 10; : : : f g :::; 4; 1; 2; 5; 8; 11; : : : f g 7Binary relations with these properties are called equivalence relations, see Section 9.10. “mcs” — 2015/5/18 — 1:43 — page 265 — #273 8.7. Remainder Arithmetic 265 according to whether their remainders on division by 3 are 0, 1, or 2. The upshot is that when arithmetic is done modulo n, there are really only n different kinds of numbers to worry about, because there are only n possible remainders. In this sense, modular arithmetic is a simplification of ordinary arithmetic. The next most useful fact about congruences is that they are preserved by addi- tion and multiplication: Lemma 8.6.4 (Congruence). If a b.mod n/ and c d.mod n/, then ⌘ ⌘ a c b d.mod n/; (8.7) C ⌘ C ac bd .mod n/: (8.8) ⌘ Proof. Let’s start with 8.7. Since a b.mod n/, we have by definition that ⌘ n .b a/ .b c/ .a c/, so j D C C a c b c.mod n/: C ⌘ C Since c d.mod n/, the same reasoning leads to ⌘ b c b d.mod n/: C ⌘ C Now transitivity (Lemma 8.6.2) gives a c b d.mod n/: C ⌘ C The proof for 8.8 is virtually identical, using the fact that if n divides .b a/, then it certainly also divides .bc ac/. ⌅ 8.7 Remainder Arithmetic The Congruence Lemma 8.6.1 says that two numbers are congruent iff their remain- ders are equal, so we can understand congruences by working out arithmetic with remainders. And if all we want is the remainder modulo n of a series of additions, multiplications, subtractions applied to some numbers, we can take remainders at every step so that the entire computation only involves number in the range Œ0::n/. “mcs” — 2015/5/18 — 1:43 — page 266 — #274 266 Chapter 8 Number Theory General Principle of Remainder Arithmetic To find the remainder on division by n of the result of a series of additions and multiplications, applied to some integers replace each integer operand by its remainder on division by n, ✏ keep each result of an addition or multiplication in the range Œ0::n/ by im- ✏ mediately replacing any result outside that range by its remainder on divi- sion by n. For example, suppose we want to find rem..444273456789 155558585555/4036666666; 36/: (8.9) C This looks really daunting if you think about computing these large powers and then taking remainders. For example, the decimal representation of 444273456789 has about 20 million digits, so we certainly don’t want to go that route. But re- membering that integer exponents specify a series of multiplications, we follow the General Principle and replace the numbers being multiplied by their remainders. Since rem.44427; 36/ 3; rem.15555858; 36/ 6, and rem.403; 36/ 7, we D D D find that (8.9) equals the remainder on division by 36 of .33456789 65555/76666666: (8.10) C That’s a little better, but 33456789 has about a million digits in its decimal represen- tation, so we still don’t want to compute that. But let’s look at the remainders of the first few powers of 3: rem.3; 36/ 3 D rem.32; 36/ 9 D rem.33; 36/ 27 D rem.34; 36/ 9: D We got a repeat of the second step, rem.32; 36/ after just two more steps. This means means that starting at 32, the sequence of remainders of successive powers of 3 will keep repeating every 2 steps. So a product of an odd number of at least three 3’s will have the same remainder on division by 36 as a product of just three 3’s. Therefore, rem.33456789; 36/ rem.33; 36/ 27: D D “mcs” — 2015/5/18 — 1:43 — page 267 — #275 8.7. Remainder Arithmetic 267 What a win! Powers of 6 are even easier because rem.62; 36/ 0, so 0’s keep repeating after D the second step. Powers of 7 repeat after six steps, but on the fifth step you get a 1, that is rem.76; 36/ 1, so (8.10) successively simplifies to be the remainders of D the following terms: .33456789 65555/76666666 C .33 62 65553/.76/1111111 C .33 0 65553/11111111 C 27: D Notice that it would be a disastrous blunder to replace an exponent by its re- mainder. The general principle applies to numbers that are operands of plus and times, whereas the exponent is a number that controls how many multiplications to perform. Watch out for this. 8.7.1 The ring Zn It’s time to be more precise about the general principle and why it works. To begin, let’s introduce the notation n for doing an addition and then immediately taking C a remainder on division by n, as specified by the general principle; likewise for multiplying: i n j rem.i j; n/; C WWD C i n j rem.ij; n/: WWD Now the General Principle is simply the repeated application of the following lemma. Lemma 8.7.1. rem.i j; n/ rem.i; n/ n rem.j; n/; (8.11) C D C rem.ij; n/ rem.i; n/ n rem.j; n/: (8.12) D Proof. By Corollary 8.6.3, i rem.i; n/ and j rem.j; n/, so by the Congru- ⌘ ⌘ ence Lemma 8.6.4 i j rem.i; n/ rem.j; n/ .mod n/: C ⌘ C By Corollary 8.6.3 again, the remainders on each side of this congruence are equal, which immediately gives (8.11). An identical proof applies to (8.12). ⌅ “mcs” — 2015/5/18 — 1:43 — page 268 — #276 268 Chapter 8 Number Theory The set of integers in the range Œ0::n/ together with the operations n and n is C referred to as Zn, the ring of integers modulo n. As a consequence of Lemma 8.7.1, the familiar rules of arithmetic hold in Zn, for example: .i n j/ n k i n .j n k/: D These subscript-n’s on arithmetic operations really clog things up, so instead we’ll just write “(Zn)” on the side to get a simpler looking equation: .i j/ k i .j k/ .Zn/: D 8 In particular, all of the following equalities are true in Zn: .i j/ k i .j k/ (associativity of ); D .i j/ k i .j k/ (associativity of ); C C D C C C 1 k k (identity for ); D 0 k k (identity for ); C D C k .
Load more

Recommended publications
  • Relations on Semigroups
    International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup
    [Show full text]
  • Modular Arithmetic
    CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on.
    [Show full text]
  • Math 327, Solutions to Hour Exam 2
    Math 327, Solutions to Hour Exam 2 1. Prove that the function f : N × N ! N defined by f(m; n) = 5m10n is one-to-one. Solution: Begin by observing that we may rewrite the formula for f as f(m; n) = 5m+n2n. Suppose f(m1; n1) = f(m2; n2), then 5m1+n1 2n1 = 5m2+n2 2n2 = k for some common integer k. By the fundamental theorem of arithmetic, the exponents in the prime decomposition of k are unique and hence m1 + n1 = m2 + n2 and n1 = n2: Substituting the second equality into the first we have m1 +n1 = m2 +n1 and hence m1 = m2. Therefore, (m1; n1) = (m2; n2), which shows that the function is one-to-one. 2a. Find all integers x, 0 ≤ x < 10 satisfying the congruence relation 5x ≡ 2 (mod 10). If no such x exists, explain why not. Solution: If the relation has an integer solution x, then there exists an integer k such that 5x − 2 = 10k, which can be rewritten as 2 = 5x − 10k = 5(x − 2k). But since x − 2k is an integer, this shows that 5 2, which is false. This contradiction shows that there are no solutions to the relation. 2b. Find all integers x, 0 ≤ x < 10 satisfying the congruence relation 7x ≡ 6 (mod 10). If no such x exists, explain why not. Solution: By Proposition 4.4.9 in the book, since 7 and 10 are relatively prime, there exists a unique solution to the relation. To find it, we use the Euclidean algorithm to see that 21 = 7(3) + 10(−2), which means that 7(3) ≡ 1(mod 10).
    [Show full text]
  • 1. Modular Arithmetic
    Modular arithmetic Divisibility Given p ositive numb ers a b if a we can write b aq r 1 for appropriate integers q r such that r a The numb er r is the remainder We say that a divides b or ajb if r and so b aq ie b factors as a times q Primes A numb er is prime if it cant b e factored as a pro duct of two numb ers greater or equal to If a numb er factors ie it is not prime then we say that it is comp osite Exercise Find all prime numb ers smaller than Here are several imp ortant questions How do we determine whether a numb er is prime How do we factor a numb er into primes How can we nd big prime numb ers How many prime numb ers are there Exercise Factor Exercise Show that every numb er is either prime or divisible by a prime numb er Theorem There are innitely many prime numbers Proof Supp ose that there are only nitely many prime numb ers P P 1 n then N P P P is not divisible by any prime hence it do es 1 2 n factor and hence it is a new prime not ˜ Divisibility tricks Recall that a numb er is even if its last digit is divisible by it is divisible by if the sum of its digits is divisible by it is divisible by if its last digit is divisible by etc Why is this Can we nd other divisibility tricks eg for One p ossible explanation Let abc b e a digit numb er so that abc a b c Then abc a b c and this is an integer only if c is an integer abc a b c and this is an integer only if c is an integer abc a b a b c and this is an integer only if a b c is an integer
    [Show full text]
  • Chinese Remainder Theorem
    THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens 1. Introduction The Chinese remainder theorem says we can uniquely solve every pair of congruences having relatively prime moduli. Theorem 1.1. Let m and n be relatively prime positive integers. For all integers a and b, the pair of congruences x ≡ a mod m; x ≡ b mod n has a solution, and this solution is uniquely determined modulo mn. What is important here is that m and n are relatively prime. There are no constraints at all on a and b. Example 1.2. The congruences x ≡ 6 mod 9 and x ≡ 4 mod 11 hold when x = 15, and more generally when x ≡ 15 mod 99, and they do not hold for other x. The modulus 99 is 9 · 11. We will prove the Chinese remainder theorem, including a version for more than two moduli, and see some ways it is applied to study congruences. 2. A proof of the Chinese remainder theorem Proof. First we show there is always a solution. Then we will show it is unique modulo mn. Existence of Solution. To show that the simultaneous congruences x ≡ a mod m; x ≡ b mod n have a common solution in Z, we give two proofs. First proof: Write the first congruence as an equation in Z, say x = a + my for some y 2 Z. Then the second congruence is the same as a + my ≡ b mod n: Subtracting a from both sides, we need to solve for y in (2.1) my ≡ b − a mod n: Since (m; n) = 1, we know m mod n is invertible.
    [Show full text]
  • Congruence Relations and Fundamental Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 75, 44545 1 (1982) Congruence Relations and Fundamental Groups YASUTAKA IHARA Department of Mathematics, Faculty of Science, University of Tok.vo, Bunkyo-ku, Tokyo 113, Japar? Communicated by J. Dieudonne’ Received April 10, 1981 1 In the arithmetic study of fundamental groups of algebraic curves over finite fields, the liftings of the Frobeniuses play an essential role. Let C be a smooth proper irreducible algebraic curve over a finite field F;I and let II (resp. n’) be the graphs on C x C of the qth (resp. q-‘th) power correspon- dences of C. In several interesting cases, C admits a lifting to characteristic 0 together with its correspondence T = 17 U P. It will be shown that in such a case the quotient of the algebraic fundamental group xi(C) modulo its normal subgroup generated by the Frobenius elements of “special points” (see below) is determined by the homomorphisms between the topological fundamental groups of the liftings of C and of T. The main result of this paper, which generalizes [2a, b] (Sect. 4), is formulated roughly as follows. Suppose that there exist two compact Riemann surfaces 31, 93’ that Yift” C, and another one, go, equipped with two finite morphisms p: 91°+ 93, o’: 93’ -+ 3’ such that 9 x @: ‘3’ --P 3 X 3’ is generically injective and (o X 4~‘) (‘3”) lifts T. Call an F,,rational point x of C special if the point (x, x9) of T (which lies on the crossing of 17 and n’) does not lift to a singular point of (u, x rp’)(l#‘).
    [Show full text]
  • Number Theory
    Number Theory Margaret M. Fleck 31 January 2011 These notes cover concepts from elementary number theory. 1 Number Theory We’ve now covered most of the basic techniques for writing proofs. So we’re going to start applying them to specific topics in mathematics, starting with number theory. Number theory is a branch of mathematics concerned with the behavior of integers. It has very important applications in cryptography and in the design of randomized algorithms. Randomization has become an increasingly important technique for creating very fast algorithms for storing and retriev- ing objects (e.g. hash tables), testing whether two objects are the same (e.g. MP3’s), and the like. Much of the underlying theory depends on facts about which numbers evenly divide one another and which numbers are prime. 2 Factors and multiples You’ve undoubtedly seen some of the basic ideas (e.g. divisibility) somewhat informally in earlier math classes. However, you may not be fully clear on what happens with special cases, e.g. zero, negative numbers. We also need clear formal definitions in order to write formal proofs. So, let’s start with 1 Definition: Suppose that a and b are integers. Then a divides b if b = an for some integer n. a is called a factor or divisor of b. b is called a multiple of a. The shorthand for a divides b is a | b. Be careful about the order. The divisor is on the left and the multiple is on the right. Some examples: • 7 | 77 • 77 6| 7 • 7 | 7 because 7 = 7 · 1 • 7 | 0 because 0 = 7 · 0, zero is divisible by any integer.
    [Show full text]
  • Bisimple Semigroups
    /-BISIMPLE SEMIGROUPS BY R. J. WARNE Let S be a semigroup and let Es denote the set of idempotents of S. As usual Es is partially ordered in the following fashion: if e,feEs, efíf if and only if ef=fe = e. Let /denote the set of all integers and let 1° denote the set of nonnegative integers. A bisimple semigroup Sis called an 7-bisimple semigroup if and only if Es is order isomorphic to 7 under the reverse of the usual order. We show that S is an 7-bisimple semigroup if and only if S^Gx Ixl, where G is a group, under the multiplication (g, a, b)(h, c, d) = (gfb-}c.chab-cfb-c.d, a,b + d-c) if b ^ c, = (fc~-\,ag<xc~''fc-b,bh,a+c-b, d) if c ^ b, where a is an endomorphism of G, a0 denoting the identity automorphism of G, and for me Io, ne I, /o,n=e> the identity of G while if m>0, fim.n = un + i"m~1un + 2am-2- ■ -un + (m.X)aun + m, where {un : ne/} is a collection of elements of G with un = e, the identity of G, if n > 0. If we let G = {e}, the one element group, in the above multiplication we obtain S=IxI under the multiplication (a, b)(c, d) = (a + c —r, b + d—r). We will denote S under this multiplication by C*, and we will call C* the extended bicyclic semigroup. C* is the union of the chain I of bicyclic semigroups C.
    [Show full text]
  • UNIVERSAL ALGEBRA Abstract Approved _____MMEMIIIMI Norman Franzen
    AN ABSTRACT OF THE THESIS OF JUNPEI SEKINO for the MASTER OF SCIENCE (Name) (Degree) ,, in MATHEMATICS presented on () t ,{, i >_: ¡ ``¡2 (Major) (Date) Title: UNIVERSAL ALGEBRA Abstract approved _____MMEMIIIMI Norman Franzen In this paper, we are concerned with the very general notion of a universal algebra. A universal algebra essentially consists of a set A together with a possibly infinite set of finitary operations on. A. Generally, these operations are related by means of equations, yield- ing different algebraic structures such as groups, groups with oper- ators, modules, rings, lattices, etc. This theory is concerned with those theorems which are common to all these various algebraic sys- tems. In particular, the fundamental isomorphism and homomorphism theorems are covered, as well as, the Jordan- Holder theorem and the Zassenhaus lemma. Furthermore, new existence proofs are given for sums and free algebras in any primitive class of universal algebras. The last part treats the theory of groups with multi- operators in a manner essentially different from that of P. J. Higgins. The ap- proach taken here generalizes the theorems on groups with operators as found in Jacobson's "Lectures in Abstract Algebra, " vol. I. The basic language of category theory is used whenever conven- ient. Universal Algebra by Junpei Sekino A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science June 1969 APPROVED: Assistant Professor of Mathematics in charge of major Acting Chairman of Department of Mathematics Dean of ra uate c ool Date thesis is presented ( ;( Typed by Clover Redfern for JUNPEI SEKINO ACKNOWLEDGMENT I wish to acknlowledge my gratitude to Mr.
    [Show full text]
  • IV.5 Arithmetic Geometry Jordan S
    i 372 IV. Branches of Mathematics where the aj,i1,...,in are indeterminates. If we write with many nice pictures and reproductions. A Scrap- g1f1 + ··· + gmfm as a polynomial in the variables book of Complex Curve Theory (American Mathemat- x1,...,xn, then all the coefficients must vanish, save ical Society, Providence, RI, 2003), by C. H. Clemens, the constant term which must equal 1. Thus we get and Complex Algebraic Curves (Cambridge University a system of linear equations in the indeterminates Press, Cambridge, 1992), by F. Kirwan, also start at an easily accessible level, but then delve more quickly into aj,i1,...,in . The solvability of systems of linear equations is well-known (with good computer implementations). advanced subjects. Thus we can decide if there is a solution with deg gj The best introduction to the techniques of algebraic 100. Of course it is possible that 100 was too small geometry is Undergraduate Algebraic Geometry (Cam- a guess, and we may have to repeat the process with bridge University Press, Cambridge, 1988), by M. Reid. larger and larger degree bounds. Will this ever end? For those wishing for a general overview, An Invitation The answer is given by the following result, which was to Algebraic Geometry (Springer, New York, 2000), by proved only recently. K. E. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves, is a good choice, while Algebraic Geometry (Springer, New Effective Nullstellensatz. Let f1,...,fm be polyno- York, 1995), by J. Harris, and Basic Algebraic Geometry, mials of degree less than or equal to d in n variables, volumes I and II (Springer, New York, 1994), by I.
    [Show full text]
  • Congruences and Modular Arithmetic
    UI Math Contest Training Modular Arithmetic Fall 2019 Congruences and Modular Arithmetic • Congruences: We say a is congruent to b modulo m, and write a ≡ b mod m , if a and b have the same remainder when divided by m, or equivalently if a − b is divisible by m. Equivalently, the congruence notation a ≡ b mod m can be thought of as a shorthand notation for the statement \there exists an integer k such that a = b + km." Here are some examples to illustrate this notation: (1) 5 ≡ 17 mod 3 (since 5 and 17 both have remainder 2 when divided by 3, or equivalently, since 17 − 5 = 12 is divisible by 3). (2) 10 ≡ −4 mod 7 (since 10 − (−4) = 14 is divisible by 7) (3) 2013 ≡ 0 mod 3 (since 2013 − 0 = 2013 is divisible by 3) (4) n ≡ 0 mod 2 if n is even, and n ≡ 1 mod 2 if n is odd. (5) n ≡ d mod 10 if n has d as last decimal digit (since then n = d + 10k for some integer k, and hence n ≡ d mod 10) • Modular arithmetic: The key fact about congruences is that congruences to the same modu- lus can be added, multiplied, and taken to a fixed positive integral power. For example, since 6 ≡ −1 mod 7, we have 61000 ≡ (−1)1000 = 1 mod 7. This type of manipulation is called modular arithmetic or congruence magic, and it allows one to quickly calculate remainders and last digits of numbers with thousands of digits. • Congruence magic: An example. Consider the problem of finding the last digit of 20132014 (which is a number with thousands of decimal digits).
    [Show full text]
  • Structures and Congruences
    Structures and Congruences Oliver Kullmann Mathematical structures Equivalence Structures and Congruences relations Subspaces and quotients Oliver Kullmann Isomorphisms theorems Computer Science Department Congruences for Swansea University groups and rings MRes Seminar Swansea, November 10, 2008 Structures and Introduction Congruences Oliver Kullmann This lecture tries to convey some key concepts from Mathematical mathematics in general, relevant for our investigations structures Equivalence into boolean algebra and Stone duality: relations First we reflect upon “mathematical structures”. Subspaces and quotients Of fundamental importance is the concept of an Isomorphisms equivalence relation “compatible” with the structure, theorems Congruences for which enables us to derive a “quotient”. groups and rings We discuss this concept in general, and especially for the algebraic structures here. Basically no proofs are given, but emphasise is put on the definitions and properties, leaving the proofs as relatively straightforward exercises. You must fill the gaps yourself!! Structures and Overview Congruences Oliver Kullmann Mathematical structures 1 Mathematical structures Equivalence relations Subspaces and quotients 2 Equivalence relations Isomorphisms theorems Congruences for 3 Subspaces and quotients groups and rings 4 Isomorphisms theorems 5 Congruences for groups and rings Structures and Three major mathematical structures Congruences Oliver Kullmann Bourbaki introduced “structuralism” to mathematics (and Mathematical the world), distinguishing
    [Show full text]